Finally, we recall that nonzero roots are simple in Yablonskii-Vorob'ev polynomials \(Q_{N}(z)\)  [1]}. In addition, nonzero roots have also been conjectured to be simple in all the Yablonskii-Vorob'ev hierarchy \(Q_{N}^{[m]}(z)\)  [2]}. Assuming this conjecture is true, then
\(\left[Q_{N}^{[m]}\right]^{\prime }(z_0)\ne 0\) . This indicates that the above leading-order asymptotics for \(\sigma _{n}(x,t)\)  never vanishes. Therefore, when \(a_{2m+1}\)  is large and \((x, t)\)  in the neighborhood of \(\left(\hat{x}_{0}, \hat{t}_{0}\right)\) , we get from (REF ) that
\(u_N(x,t) = \frac{\sigma _{1}}{\sigma _{0}}e^{\textrm {i}t} =e^{\textrm {i}t}\left(1- \frac{4[1+2\textrm {i}(t-\hat{t}_{0})]}{1+4(x-\hat{x}_{0})^2+4(t-\hat{t}_{0})^2}\right) + O\left(a_{2m+1}^{-1/(2m+1)}\right),\) 
